# Set in Discrete Topology is Clopen

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## Theorem

Let $T = \struct {S, \tau}$ be a discrete topological space.

## Proof

Let $U \subseteq S$.

By definition of discrete topological space, $U \in \tau$.

By definition of closed set, $\relcomp S U$ is closed in $T$, where $\relcomp S U$ is the relative complement of $U$ in $S$.

But from Set Difference is Subset:

- $\relcomp S U = S \setminus U \subseteq S$

and so:

- $\relcomp S U \in \tau$

That is, $\relcomp S U$ is both closed and open in $T$.

Then by Relative Complement of Relative Complement:

- $\relcomp S {\relcomp S U} = U$

which is seen to be both closed and open in $T$.

$\blacksquare$

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 3.2$: Topological Spaces: Exercise $6$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$: Discrete Topology